Tennis serves are usually aimed to the receiver’s left or right. (Center serves are unusual, at least in championship play.) Since the server needs to keep the receiver guessing, rational play dictates a mixed strategy. The best mixture will depend upon many factors: whether the players are right-handed or left-handed, possible weaknesses of forehands or backhands, individual peculiarities of play, the current point score, the direction of the sun, possible referee bias, and more.

Despite these complications, the test of an optimal mixed strategy is that all the pure strategies being played must on average be equally profitable. (If they were not, it would pay to choose the more profitable option more often.) In particular, for the player with the service, serves to the left and serves to the right should have equal success rates. Mark Walker and John Wooders obtained data on all first serves in 10 important professional tennis matches  most of them final championship matches.a If the players were choosing rationally, in a given match there might be a large disparity between the percentages of left and right serves, but left and right serves should have been, on average, equally likely to win points.

The results reported here refer to the service choices of the ultimate match winner when the score was at “deuce.” The left-right mixtures are percentages that sum to 1, since center serves (only about 6%) were not counted. The win rates for both left serves and right serves are all well above 50%, reflecting the advantage at tennis of having the serve. (Winning was defined here as gaining the point, whether earned on the initial service as an ace or only after additional strokes.) The imbalance between left and right service proportions was sometimes very great. Rosewall at Wimbledon in 1974 served left 93% of the time. That was the extreme, but the percent differences between the proportions of left and right serves were generally quite large  averaging about 39%. In contrast, the win rates for the two types of service were close together, diverging on average by only around 10%.

The authors’ interpretation was that, unconsciously perhaps, championship tennis players appreciate the need to mix their strategy choices, and do so in a way close to the theoretical optimum.

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aMark Walker and John Wooders, “Minimax Play at Wimbledon,” American Economic Review,v.91 (December 2001).